By Valentino Magnani

We receive an intrinsic Blow-up Theorem for normal hypersurfaces on graded nilpotent teams. This process permits us to symbolize explicitly the Riemannian floor degree by way of the round Hausdorff degree with recognize to an intrinsic distance of the gang, specifically homogeneous distance. We practice this consequence to get a model of the Riemannian coarea forumula on sub-Riemannian teams, that may be expressed when it comes to arbitrary homogeneous distances.We introduce the common type of horizontal isometries in sub-Riemannian teams, giving examples of rotational invariant homogeneous distances and rotational teams, the place the coarea formulation takes an easier shape. by way of an identical Blow-up Theorem we receive an optimum estimate for the Hausdorff measurement of the attribute set relative to C1,1 hypersurfaces in 2-step teams and we end up that it has finite Q − 2 Hausdorff degree, the place Q is the homogeneous measurement of the crowd.

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7) THEOREM. Let G = (x, y :xP = l,yQ = l,yxy-1 = x'), where p is an odd prime, q any divisor of p - 1, and r is a primitive q-th root of 1 mod p. Let w be a primitive p-th root of lover Q, and let L be the unique subfield of Q(w) such that (Q(w): L) = q. Put S = alg. int. {L}. Let H = (y), a cyclic group of order q. Then there is a surjection Cl ZG ~ Cl S Ea Cl ZH 30 IRVING REINER whose kernel Do is a cyclic group of order q/(q, 2). Further, the sequence o -Do -D(le) -D(ZH) - 0 is exact. D(le) I = 1 if e is a dihedral group of order 2p.

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